Period spaces for pdivisible groups urs hartl university of m. A classi cation of pdivisible groups over o k 24 6. We prove several results about pdivisible groups and rapoportzink spaces. This construction is based on the moduli theory of pdivisible groups of a fixed isogeny type. Talk 2 last time we defined divisible groups and looked at some basic examples and constructions. This is a set of notes for a course we gave in the second week of august in the 2006 cmi summer school at go. Combine the discussion above, to any pair, b where is a one parameter subgroup. In a nutshell, a pdivisible group x s over a base scheme s in equi.
We determine their dimensions and their sets of connected components and of irreducible components. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in. Patrick walls dieudonn e modules and p divisible groups. We study the global structure of moduli spaces of quasiisogenies of pdivisible groups introduced by rapoport and zink. Tate 1967 defined a pdivisible group of height h over a scheme s to be an inductive system of groups g n for n. Using the concept of rigidanalytic period maps the relation of p adic period domains to moduli space of p divisible groups is investigated. We study the global structure of moduli spaces of quasiisogenies of polarized pdivisible groups introduced by rapoport and zink. Consider all the strings of length math p math, where each position of the string can be occupied by one of mathamath distinct alphabets. This write up is devoted to some basic but important aspects of the structure theory of pdivisible groups.
There is a deep correspondence between the morava etheory of spaces and the algebraic geometry of the formal group associated to e n. Introduction padic symmetric domains quasiisogenies of pdivisible groups moduli spaces of pdivisible groups appendix. Lastly, we combine the results in the etale case and. On the other hand, reading this book is really indispensable if you want to master pdivisible groups.
On the other hand, reading this book is really indispensable if you want to master p divisible groups. The 3 contains some theorems about the action of gal k. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian. Fp, the moduli space of polarized abelian varieties of dimension g in positive. In this paper we compute the e ect of the transchromatic generalized characters of 7 on the morava etheory of symmetric groups. Lectures on pdivisible groups lecture notes in mathematics. The 3 contains some theorems about the action ofgalkjk on the completion c ofthe algebraic closure k of a.
For any pdivisible group g over k, there exists a unique newton polygon. X 2pdoes in general not imply that x 1 and x 2 are isomorphic. Families of pdivisible groups with constant newton polygon. Associate professor kartik prasanna, cochair professor brian d. An abelian group is divisible if and only if it is p divisible for every prime p. Tate modules, pdivisible groups, and the fundamental group.
In this subsection we will combine the considerations in 8. Subgroups of pdivisible groups and centralizers in. If g ap1 for an abelian variety a, then the newton polygon. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p adic hodge theory. April 25, 2008 abstract in their book rapoport and zink constructed rigid analytic period spaces for fontaines.
The system is then a divisible group of height and is called the serre dual to the divisible group. The global structure of moduli spaces of polarized. Over padic nagata rings, formal pdivisible groups are classified by nilpotent displays according to t. The space of level 0, and the period morphism 44 6. We saw that the divisible group of an abelian variety over a perfect field of positive characteristic encodes similar data to. It can be shown that a nite group is nilpotent if and only if it possesses a central series. Let x be a p divisible group with constant newton polygon over a normal noetherian scheme s. In their book rapoport and zink constructed rigid analytic period spaces for fontaines filtered isocrystals, and period morphisms from moduli spaces of pdivisible groups to some of these period spaces. Period spaces for pdivisible groups am141 by michael rapoport, thomas zink pdf, epub ebook d0wnl0ad in this monograph p adic period domains are associated to arbitrary reductive groups. Annals of mathematics, 161 2005, 10211036 minimal pdivisible groups by frans oort introduction a pdivisible group x can be seen as a tower of building blocks, each of which is isomorphic to the same. On specializations of minimal pdivisible groups nobuhiro higuchi and shushi harashitay august 2, 2016 abstract in this paper, for any pair of newton polygons with. If the isocrystals of the pdivisible groups are simple, we compute the cohomology of the moduli space. The multiplication by p on a pdivisible group is obviously an isogeny. The basics of the theory are welldeveloped in the first few chapters, but the last 2 chapters are too short.
If x is an abelian variety over the complex numbers, the associated analytic manifold can be described as a complex torus v. We study pdivisible groups xover a base scheme sand, colloquially, a pdivisible group over a base scheme of positive dimension will be called a \family of pdivisible groups, such that the newton polygon of a. Newton polygong shushi harashita kobe estimate the np of a pdivisible group 24. Moduli spaces of pdivisible groups and period morphisms. H where g and h are pdivisible groups on s is a morphism of pdivisible groups if f is a morphism of f. The serretate theorem provides a converse in the case of in. We prove that there exists an isogeny to x y such that y admits a slope filltration. Estimating the newton polygon of a pdivisible group from its.
Period spaces for pdivisible groups am141, volume 141. The most exciting thing discused will be the relation between connected pdivisible groups and divisible formal lie groups over r. Lectures on pdivisible group 3 is an exact sequence in ab. Let k be the fraction eld of a complete discrete valuation ring with mixed characteristic 0. We determine the image of these period morphisms, thereby contributing to a question of grothendieck. The restriction of the functor bt to the category of displays is faithful.
For both abelian schemes and pdivisible groups, we have the notion of deformations. I spent the period apriljune 2010 at the university of pennsylvania and worked. Unlike the definition of minimal pdivisible groups in 7, which is a specific construction, our definition of minimal f crystals is all those f crystals whose isomorphism numbers are. It is fully faithful, if the ideal of nilpotent elements in ris a nilpotent ideal.
The theory of pdivisible groups plays an important role in arithmetic. Historically, results and conjectures surrounding pdivisible groups were the main stimulus for the development of padic hodge theory, and they continue to be relevant in modern research. In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. On period spaces for pdivisible groups request pdf. Using the concept of rigidanalytic period maps the relation of padic period domains to moduli space of pdivisible groups is investigated. Subgroups of pdivisible groups and centralizers in symmetric.
Jared weinstein, the geometry of lubitate spaces, lecture 1. Displays and formal p divisible groups springerlink. The following main theorem gives the comparison of our theory and the crystalline dieudonn e theory of grothendieck and messing. It follows that for pdivisible groups, the group hom sg. The purpose of this course was to carefully lay the foundations for. Using the corresponding results for nonpolarized pdivisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components. Theorem the witt group scheme w n of length n is an injective cogenerator in a n.
Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way. But avoid asking for help, clarification, or responding to other answers. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of padic hodge theory. Let g k galkk, so g k acts on c k be isometries in the evident manner. To each barsottitate group g, we can associate a dieudonn.
I will explain why moduli spaces of pdivisible groups become perfectoid spaces at infinite level. In the paper groupes pdivisible sur les corps locaux fon2. The proof uses the grothendieckillusie deformation theory of truncated pdivisible groups. Are there anypdivisiblegroups over z other than products ofpowers ofgmp and ofqplzp1 2. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. Numerous and frequentlyupdated resource results are available from this search. A morphism between pdivisible groups is a collection of morphisms f.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a. In their book rapoport and zink constructed rigid analytic period spaces for fontaines filtered isocrystals, and period morphisms from moduli spaces of p divisible groups to some of these period spaces. Minimal pdivisible groups by frans oort introduction a pdivisible group x can be seen as a tower of building blocks, each of which is isomorphic to the same. Subgroups of pdivisible groups and centralizers in symmetric groups nathaniel stapleton november 8, 20. K on the completion c of the algebraic closure of a local field k of characteristic 0. The moduli spaces obtained in this way are formal schemes over the. In order to provide an algebrogeometric description of the answer we must develop the relationship between transfers in e n and transfers for the cohomology theory c t constructed in. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. In this paper we compute the e ect of the transchromatic. What is a proof that apa is divisible by p where p is a. Zink princeton university press princeton, new jersey 1996.
Part ii brandon levin this talk discusses the main results of tates paper pdivisible groups 6. Normal forms of lattice chains the formal hecke correspondences the period morphism and the rigidanalytic coverings padic uniformization of shimura varieties. Thanks for contributing an answer to mathematics stack exchange. Pis a formal p divisible group of height equal to rank rp. We call xa pdivisible group with constant newton polygon. We determine the image of these period morphisms, thereby answering a question of grothendieck.
A pdivisible group g over r of height h is an inductive system g g. A pdivisible group x can be seen as a tower of building blocks, each of which is isomorphic to the same finite group scheme xp. Cm pdivisible groups over finite fields by xinyun sun a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2011 doctoral committee. T is a morphism, a deformation of a s is an abelian scheme a0 t together with isomorphisms a 0 t t s. The crystals associated to barsottitate groups 18 4. Statement of purpose the aim here is simply to provide some details to some of the proofs in tates paper t. On period spaces for pdivisible groups sciencedirect. Therefore all the pdivisible groups on a base scheme s form a category denoted. Oct 19, 2014 there is a very beautiful combinatorial argument that proves the above proposition. We give examples showing that only in rare cases the image is all of the rapoportzink period space. Exterior powers of barsottitate groups people eth zurich. Hn are morphisms of group schemes and compatible with the transition maps. Nontrivial divisible groups are not finitely generated. From the point of view of padic hodge theory, this is a foundational paper and within this setting, much of the.
Let kk be an algebraic closure and c k its completion. Relations with formal lie groups in this section we assume r complete, noetherian, local, with residue field k ofcharacteristicp o. Remark what is additionally interesting about example is that it provides an injective cogenerator for the category ab of abelian groups. Hoaran wang, moduli spaces of pdivisible groups and period morphisms, masters thesis, 2009, pdf. Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. We prove several results about p divisible groups and rapoportzink spaces. For ourpresentpurposes, an ndimensional formal lie group rover r can bedefined as a suitable. As a rst example of formal groups, we want to show that given a commutative.
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